Make it clear to your students from the beginning what is expected of them. I try to show them that there's a certain social contract involved, at least academically: they have the freedom to learn at their own pace, but with that freedom comes responsibility. It's up to them to decide when they've learned something thoroughly, and when they can skim through a subject.
Practically, I like to give my students a lot of freedom: they can study how they want, with whom they want, and where (in the classroom) they want, as long as their work gets done and they're not disturbing other people.
I think it's perfectly okay to joke around with the kids and answer personal questions (provided you want to, that is), but be aware of when you do it. You don't want to get them all riled up in the beginning of class, for example, because then they'll have trouble settling down for the rest of the class session. Don't tell them anything you wouldn't want to be repeated among the whole student body, though, because if it's juicy enough, it will be. Likewise, I would refrain from repeating any gossip at all about another staff member, unless you're positive they wouldn't mind for the entire student body to know.
Don't be afraid to admit you've made a mistake, or to apologize to a student. I try to apologize to students when I've lost my temper with them, and explain why I did. Making a mistake and then trying to cover it up is much worse than making a mistake, admitting it, and rectifying the situation. You don't have to be omniscient, but you do have a responsibilty to exercise good judgment.
I like passing out an intial questionnaire to ask the students about their motivations for being in the class, as well as to find out more about their personalities. You can learn a lot about a student from their response to the question "Why did you decide to take precalculus math?" If their answer is "Because my father wanted me to", it might explain why they have low motivation- or, why they work as feverishly as if the hounds of hell are after them.
I think it's a good idea to try to keep an eye on class dynamics. Personality conflicts between kids are natural; it's your job to make sure they don't get out of hand. For example, I had a class where most of the class kept telling two kids, "Shut up!" I told them that if they were going to ask someone to quiet, they'd have to be nice about it, which got them saying "Shut up, please!" instead. I think it was a fairly good compromise: they moderated their tone somewhat and weren't so harsh about it. You have to respect that the students will need to work out these problems among themselves. They can and will, they just might need a little guidance.
Remember that at this age, what the students need to learn, as much as (or even more than) the knowledge is the ability to learn. Preferably, the ability to teach themselves. A lot of them are going to have to teach themselves because they won't be taught in their home schools. The summer program is one of the few times they've been pushed to their limits, to really understand how much they can learn and what they can do. It's important that it be an affirming experience for them. Each student will have their own pace, and their own tough spots or easy spots. Just because they're having difficulty with one topic doesn't mean they can't finish the course. You need to know this so that they can know this. Be positive and sincere when you're encouraging a student. They've had enough bad and insincere teachers in the past to know when you're faking it. They'll do better if they know they can trust you.
Also remember that each student has their own style of learning. Some are loners, some like to work with a buddy, some in small groups. If you have a group of students all at the same topic, and they're having difficulty, give a lecture. Get them to come up to the board and copy problems and solve them. Sometimes they just need a change of pace, or a fresh perspective on the problem. There are some subjects where you'll almost always lecture: in Algebra II, for example, it's Conics.
Don't be afraid to make up your own quizzes or worksheets if you think what's in the book is complete bunk. Sometimes it is.
If a student asks a question you can't answer right away, and you can't solve it within 3-4 minutes, admit it and tell them you'll work on it and get back to them. This applies to any extra topics they might want to learn, too. This goes back to being honest with students: they won't disrespect you for not knowing all the answers immediately, but they will for lying to them. It's fine for you to say "I don't see the answer right now, but let me work on it and we'll go over it together later."
I like to do extra problems for the last half hour or so. Students get antsy around then anyways. You can plan ahead, or do what I call Extra Problems Improv, which is asking the students for ''brainteasers'' and logic puzzles. Most of them will know one or two, and there might be some you've never seen before.
I favor the incremental freedom method of teaching. Well, I don't know if that's a good name for it. What I mean is this: I never raise my voice during the first week, ever. Not during the second week either if I can help it. If you start out by raising your voice, you'll have nowhere to go and eventually the students will not respond when you shout at them to be quiet. It's better, I think, to be fairly relaxed, that way if they get out of line, raising your voice really has an effect. In the same way, I try to keep the breaks fairly short until about half-way through the session; ditto with going outside for class. If you allow too much freedom in the beginning of the session, the class will just want more until it becomes anarchy. This sounds cynical, but practically speaking, it works. In the beginning of the session, the students will be better able to concentrate and won't need as much break time.
Some of the clues are: inability to stay in their seats, drastic mood swings, very short attention span, need for a lot of attention, extreme forgetfulness in practical matters. You don't want to ostracize these kids by drawing attention to their behavior (in most cases, they already have been ostracized to a certain extent by their peers), but you want to work with them to overcome their problems.
If you know a student has been diagnosed with ADD, check with the RA to see if they (the student, not the RA) are on some sort of medication, and if they are, are they taking it regularly. In the classroom, try to structure things as much as possible for them. Try setting goals of different lengths: for example, instead of saying "study sections 4.6-4.8", assign problems in those sections and have the student hand them in.
As much as possible, don't let their ADD be an excuse for bad behavior. You aren't doing the student any favors by allowing them to act up or giving them special dispensation; they won't get it from their home schools either.
In spite of all this, students frequently get really worked up about it and have been known to burst into tears when they don't cert. Be prepared for this; despite your best efforts, most of these students may get kind of wonky about it simply have never encountered a situation where they have done less than the top 10%.
I generally make a chart with headings like: Student name, course, pace, thoroughness, written work (proofs, quizzes, finals, etc.) work in class (meaning how easily they were distracted, how they worked with others), certification, recommendation. Usually a few words in each of these fields will be a good reminder for what you want to write. Remember that you can start writing evals before a student finishes a course, just leave notes to yourself about things to fill in.
A lot of evals will be similar; that's okay if the students aren't best buddies from home and won't be reading each other's evals. In most cases, they won't be.
Get your TA to read over the evals for both content and grammar as soon as you have rough drafts done.
Don't be afraid to write the truth in evals, but don't be unnecessarily harsh. If you write something that is negative, your academic dean may try to call you on it and get you to change it. If you believe that it's truthful, and that it won't be any surprise to the student (i.e. "you were a frequent disruption in class" and the student knows you mean that he wouldn't stop throwing his shoes across the room), stand up for it. You shouldn't have to compromise what you see as the truth for the sake of having a ''nice'' sounding eval.
I also write course evals and staff evals. Staff evals includes all TAs and RAs with whom I've worked. Course evals means anything from logistics ("don't put another precalc class in the room underneath the physics lab!") to suggestions for new directions. It is unclear if these evals are actually read by anyone other than ourselves and our academic deans; but I still write them on the premise that the only way I can be sure they aren't read is if I don't write them.
Try to stifle any impulses you may have to sock a parent, even if you know they've been cruel to their child. If a child suffers from low self-esteem, and you know the parent is a major factor, you can try to convince the parent otherwise, but there is very little you can say in ten minutes that will change a parent's attitude. This is the sad truth.
I would advise trying to bolster the student's self-esteem yourself; let them know that you believe in them and that they can succeed. Their presence in the CTY program should be evidence enough of their intelligence; being smart people, they should know this, but you'd be surprised. I never expect that encouragement over three weeks is going to change their outlook on life, but one can hope.
Encourage them to give lectures, be it on special topics or mundane stuff like rational expressions. Encourage your TA to help with daily discipline, and writing evals. Remember that most TAs aspire to being instructors someday, and they need experience in order to get good at things. If you take all the interesting and/or difficult work on yourself, you aren't doing them any favors.
Sometimes students like having you visit them on their halls. Make sure you ask the RA first and let them know you're coming to visit; in general, it's a good thing and pretty fun. You'll probably be surprised at how loud the kids can be outside of class.
Some RAs are very conscientious, some aren't. It isn't worth getting really worked up about, unless you have a student who is in trouble and the RA is unresponsive. Then you need to get on the RA's case about stuff. Take it to the SRA if you feel it's necessary.
Don't get involved romantically with any student. It's great to be their pal, but if you think they may be getting any ideas about romance, you should put some distance between you. Besides being a serious offense in the eyes of CTY administration (and a good way to get fired), quite simply, it's a bad idea. It's a conflict of interest in the worst way, and besides which, no matter how mature a student seems, the fact is that s/he is not of the age to get involved with an adult (which is what you presumably are). For some of us, this rule is not a problem. However, students have been known to glom on to certain staff members as objects of adolescent desire. Remember this, and as basic CYA policy, talk to someone on the staff (subject coordinator, academic dean, counselor, whomever you feel you can trust to be objective) if you feel a situation is developing.
I think it's a good idea to be as friendly as possible to non-math staff members. Some of them won't want to be friendly to you, but sinking to their level of manners doesn't help you or your image any. Try inviting them along to Thundering Herd activities; who knows, they may enjoy themselves.